let n be Nat; :: thesis: for V being RealLinearSpace
for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
degree Ka = degree (BCS (n,Ka))
let V be RealLinearSpace; :: thesis: for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
degree Ka = degree (BCS (n,Ka))
let Ka be non void affinely-independent SimplicialComplex of V; :: thesis: ( |.Ka.| c= [#] Ka implies degree Ka = degree (BCS (n,Ka)) )
defpred S₁[ Nat] means ( degree Ka = degree (BCS ($1,Ka)) & not BCS ($1,Ka) is void & BCS ($1,Ka) is affinely-independent );
assume A1: |.Ka.| c= [#] Ka ; :: thesis: degree Ka = degree (BCS (n,Ka))
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: S₁[n] ; :: thesis: S₁[n + 1]
A4: [#] (BCS (n,Ka)) = [#] Ka by A1, Th18;
( BCS ((n + 1),Ka) = BCS (BCS (n,Ka)) & |.(BCS (n,Ka)).| = |.Ka.| ) by A1, Th10, Th20;
hence S₁[n + 1] by A1, A3, A4, Th28, Th31; :: thesis: verum

end;

A5: S₁[ 0 ] by A1, Th16;
for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A2);
hence degree Ka = degree (BCS (n,Ka)) ; :: thesis: verum